An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

Authors


Abstract

We propose an original scheme for the time discretization of a triphasic Cahn–Hilliard/Navier–Stokes model. This scheme allows an uncoupled resolution of the discrete Cahn–Hilliard and Navier-Stokes system, which is unconditionally stable and preserves, at the discrete level, the main properties of the continuous model. The existence of discrete solutions is proved, and a convergence study is performed in the case where the densities of the three phases are the same. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq. 2013

I. INTRODUCTION

The complexity of multiphasic flows basically lies in the fact that the time evolution of interfaces, whose position is an unknown of the problem, may lead to their deformation, their break-up or coalescence. Moreover, interfaces obey to physical phenomena where capillary effects play an important role.

The various domains of application, where multiphasic flows are involved, are generally complex; the experimentation and measurements are quite difficult, onerous, and most often not very accurate. For instance, in nuclear safety [1], the understanding of interaction between molten corium (lava-like molten mixture of portions of nuclear reactor core) and concrete (last confinement barrier) is a major issue. An approach using direct numerical simulations allows to access to instantaneous quantities at each point of the flows.

Due to their ability to capture interfaces implicitly, diffuse interfaces models are attractive for the numerical simulations of multiphase flows. In this article, we consider a model that couples the Cahn–Hilliard system and the Navier–Stokes equation.

A. The Cahn–Hilliard Model

In diffuse interfaces theory, the interfaces are assumed to have a nonzero thickness ε (which is here a constant parameter of the model). Interfaces are considered as mixing areas, and the phase i can be represented by a smooth phase indicator ci called “order parameter” (which may be understood here as the volumic fraction of the phase i). Thus, the system contains as many unknowns ci as phases. These unknowns vary between 0 and 1 (values that correspond to pure phases by convention) and are linked by the relationship equation image.

A complete derivation of this kind of models for diphasic flows is presented in Refs. [2–5]. Different extensions have been proposed for the simulations of three-phase flows in Refs. [6–8]. We consider in this article the triphasic Cahn–Hilliard model taken from Ref. [6]:

equation image(1.1)

where M0(c) is a diffusion coefficient which is called “mobility” and may depend on c = (c1, c2, c3). The functions fmath image are defined by:

equation image

where ΣT is given by equation image. This system is a gradient flow for the following energy functional under the constraint of volume conservation:

equation image(1.2)

where Ω denotes an open, bounded, connected, and smooth domain of equation image (d = 2 or 3). The “intermediate” unknowns μi, called “chemical potentials,” are the functional derivatives of the triphasic Cahn–Hilliard energy (1.2). The rather intricate expression of fmath image let us ensure the constraint:

equation image

We introduce the hyperplane equation image of equation image, to simplify notation in the sequel.

The expressions of the triphasic Cahn–Hilliard potential F and the constant triplet Σ = (Σ1, Σ2, Σ3) was derived in Ref. [6], so that the model can correctly take into account the surface tensions values σ12, σ13, and σ23 prescribed between the different pairs of phases, and it is consistent with the two-phase situations: the triphasic model has to exactly reproduce diphasic situations when one of the three phases is not present. The coefficient Σi is given as a function of the surface tensions:

equation image(1.3)

and the triphasic potential F has the following form:

equation image

where Λ is an arbitrary smooth function of c.

Note that, in the sequel, we do not assume that the coefficients Σi are non-negative, so that the model can handle some total spreading situations. However, as it is proved in Ref. [6], the following condition is necessary to ensure the well posedness of the system:

equation image(1.4)

This condition is equivalent to the coercivity of capillary terms and ensures that these terms bring a positive contribution to the free energy equation image. This is detailed in the following proposition:

Proposition 1.1 ([6, Prop 2.1]). Let equation image. There exists Σ > 0 such that, for all n ≥ 1, for all equation image such that ξ1 + ξ2 + ξ3 = 0,

equation image

if and only if the two following conditions are satisfied:

equation image(1.5)

This proposition will be useful in the sequel.

Remark 1. Owing to (1.3), the second part of condition (1.5) is always satisfied and consequently it is sufficient to assume that the condition (1.4) holds, for applying Proposition 1.1.

The existence of weak solutions for problem (1.1) together with initial and Neumann boundary conditions (for order parameters ci and chemical potentials μi) was proved in Ref. [6] (see Ref. [9] for an alternative proof based on numerical schemes) in 2D and 3D under the following general assumptions:

  • The mobility M0 is a bounded function of equation image class, and there exists three positive constants M1, M2, and M3 such that:

    equation image(1.6)
  • The Cahn–Hilliard potential F is a positive function of equation image class that satisfies the following assumptions of polynomial growth: there exist a constant B1 > 0 and a real p such that 2 ≤ p < +∞ for d = 2 or 2 ≤ p ≤ 6 for d = 3, and

    equation image(1.7)

In the previous items, the notation D stands for the differentiation operator.

B. Coupling with Hydrodynamic

The coupling between the Cahn–Hilliard and Navier–Stokes systems is obtained by:

  • 1adding a transport term u · ∇ci in the evolution equation of each order parameter ci, (i ∈ {1,2,3}), which is the first equation of system (1.1).
  • 2defining the density and viscosity as smooth functions of order parameters c.
  • 3adding a capillary forces term equation image in the right-hand side of the momentum balance (in the Navier–Stokes equation).

Furthermore, we adopt a particular form of the Navier–Stokes equation, initially proposed in Ref. [10] (see also Refs. [5, 11]), which ensures an energy balance without using the equation of mass conservation. It relies on the following inequality:

equation image

the domain Ωt being an arbitrary bounded smooth domain moving at the fluid velocity u [12].

Hence, the triphasic Cahn–Hilliard/Navier–Stokes, we study here, is constituted with following equations:

equation image(1.8)

where the vector g stands for the gravity; the tensor Du stands for the symmetric gradient of the velocity u; the scalar p stands for the pressure; the density and viscosity are defined by:

equation image

where ϱ1 (resp. ϱ2, ϱ3) and η1 (resp. η2, η3) are the prescribed values (assumed to be constant) in phase 1 (resp. 2, resp. 3) and the function hλ (λ = 0.5) is defined by:

equation image

We supplement this system with Neumann boundary conditions for order parameters ci and chemical potentials μi, that is, for i = 1,2,3,

equation image(1.9)

and with a homogeneous Dirichlet boundary condition for the velocity, that is,

equation image(1.10)

Owing to these boundary conditions (1.9) and (1.10), we introduce the following function spaces:

equation image

Finally, we assume that the following initial condition holds:

equation image(1.11)

where equation image and equation image are given.

C. Outline of the Article

In Section II, we describe the time and space discretization of the problem (1.8). We then prove, in Section III, the unconditional stability of the scheme and the existence of approximate solutions. Section IV is devoted to numerical experiments. In the last section (Section V), we prove the convergence of approximate solutions toward weak solutions of the Cahn–Hilliard/Navier–Stokes system in the case where the three fluids have the same density. In particular, we prove the following existence theorem by passing to the limit in the numerical scheme in Section V:

Theorem 1.2 (Existence of weak solutions in the homogeneous case). Assume the coefficients1, Σ2, Σ3) satisfy the condition (1.4), the mobility satisfy (1.6), and that the Cahn–Hilliard potential F satisfy the condition (A). Assume the densities of the three fluids are equal, that is, ϱ1 = ϱ2 = ϱ3 = ϱ0, equation image. Consider the problem (1.8) together with initial condition (1.11) and boundary conditions (1.9) and (1.10). Then, for all tf ∈]0, +∞[, there exists a weak solution (c, μ, u, p) on [0, tf[ such that

equation image

II. DISCRETIZATION OF THE CAHN–HILLIARD/NAVIER–STOKES MODEL

A. Time Discretization

Let equation image. The time domain [0, tf] is uniformly discretized with a fixed time step equation image; we define tn = nΔt, for all n ∈ [[0; N]]. We assume that the functions equation image and equation image (n ∈ [[0;N − 1]]) are given and we describe the system we have to solve to compute the unknowns equation image and equation image at time tn+1.

We first describe, in two distinct paragraphs, the schemes we use to discretize the Cahn–Hilliard and Navier–Stokes equations without considering the coupling terms. We then explain, in the next two paragraphs, the reasoning leading to the discretization of the coupling terms before writing the complete scheme in the last paragraph. For more details on the time discretizations of the triphasic Cahn–Hilliard model, the reader may refer to the article [9] (and references therein). Several articles in the literature are devoted to the study of discretizations of the Navier–Stokes equation: we refer in particular to the articles [10, 13], which deal with variable density models.

Cahn–Hilliard System.

We consider a time discretization of the Cahn–Hilliard system of the form: for i = 1,2,3,

equation image

where cmath image = (1 − β)cmath image + β cmath image, β ∈ [0.5, 1], and Mmath image = M0((1 − α)cncn+1), α ∈ [0, 1]; the discretization of transport terms is postponed to “Coupling Terms” “Coupling Terms” section.

This kind of discretizations was presented and studied in Ref. [9]. This is out the scope of the article. We assume that the discretization Dmath image(cn, cn+1) of the term fmath image is of the form:

equation image

where dmath image stands for a semi-implicit discretization of ∂iF. We assume that the two following basic properties hold for all i ∈{1, 2, 3}:

equation image(2.1)
equation image(2.2)

the notation D means here the derivative of dmath image with respect to the second variable. The assumption (2.1) is a consistency assumption and the assumption (2.2) is the counterpart of the polynomial growth assumption (1.7) on F. Many possible choices for the discretization of the term dmath image were presented in Ref. [9]. For instance, we consider in numerical tests of this article (Section IV) the following expression:

equation image

This scheme was built to ensure the following equality:

equation image

and consequently a discrete energy equality which is obtained by multiplying the first equation of the Cahn–Hilliard system by μmath image, the second one by cmath imagecmath image, writing the equality of left-hand sides and summing for i = 1,2,3.

Navier–Stokes System.

We now present the time discretization of the momentum balance of the Navier–Stokes system:

equation image

We separately present the discretization of the different terms (1)–(3) involved in the above equation; for each of them, we give their contribution to the energy balance obtained at the discrete level by multiplying the equation by un+1 and integrating on Ω.

  1. Term (1): Using the formal equality equation image, we choose the following discretization of Term (1) (see Ref. [13]):

    equation image

    Its contribution to the energy balance is:

    equation image
  2. Term (2): The Term (2) is linearized using an explicit velocity for the transport:

    equation image

    Its contribution to the energy balance vanishes. Indeed, for all equation image, we have:

    equation image

    In particular, when we take νu = un+1, the above term vanishes.

  3. Term (3): We discretize the Term (3) with an implicit scheme: −div (2ηn+1Dun+1). Its contribution to the energy balance is equation image.

    Thus, we adopt the following discretization of the Navier–Stokes equation:

    equation image

The discretization of the capillary forces term is described in the next paragraph.

Coupling Terms.

We give in this paragraph the discretization of coupling terms. That is the transport terms u Δ∇ci in the Cahn–Hilliard equations, and the capillary forces term equation image in the momentum balance (Navier–Stokes equation). At the continuous level, when writing the energy balance, the contributions of these two terms counterbalance each other. At the discrete level, we saw that the energy balance is obtained, for the Cahn–Hilliard system, by multiplying the transport terms by μmath image before summing up for i = 1,2,3 and, for the Navier–Stokes equation, by multiplying the capillary forces term by un+1.

Consequently, it is easy to see that when all the terms mentioned above are discretized with an implicit scheme (cf Ref. [14] for the diphasic case), that is, un+1 · ∇cmath image and equation image, the balance is also true at the discrete level. However, this discretization introduces a strong coupling between the Cahn–Hilliard and Navier–Stokes systems. The discrete system is difficult to solve in practice.

It is possible to uncouple the two systems (cf Ref. [15] for the diphasic case, Ref. [11] for the triphasic case) using an explicit velocity (i.e., the velocity at time tn) in the Cahn–Hilliard equation: un · ∇cmath image. However, the contributions of the transport terms in the Cahn–Hilliard system and the contribution of the capillary forces in the Navier–Stokes equation do not counterbalance when writing the discrete energy balance, which contains the additional term: equation image. It is difficult to attribute a sign to this term, and the scheme stability is obtained only conditionally (cf Ref. [15], assuming for instance that the ratio between the time step and the mesh size is bounded).

We first observe that it is possible to uncouple the resolution of the Navier–Stokes system and the taking into account of capillary forces. The taking into account of the capillary forces is performed during a first step which provides an intermediate velocity u* which is then used in the Cahn–Hilliard system. The Navier–Stokes system is then solved in a second step. The scheme reads:

  • i.Taking into account of capillary forces:
    equation image
  • ii.Cahn–Hilliard system:
    equation image
  • iii.Navier–Stokes system:
    equation image

This discretization is unconditionally stable but the system of step (i) (Darcy problem) is still coupled with the Cahn–Hilliard equations (system (ii)).

We propose to forget for a moment the divergence free constraint imposed to u* (and consequently the associated pressure term ∇p*) in the system of the step (i). This leads to define u* as follows:

equation image

This definition of u* is explicit, and u* can be replaced by its expression in the Cahn–Hilliard system thus eliminating the coupling with the Navier–Stokes equation.

The problem is that u* is not divergence free. Nevertheless, note that the divergence of u* is of order Ot) and that the property u* · n = 0 is still satisfied on Γ. Now, the question is: is it possible to discretize the transport term in the Cahn–Hilliard equation to preserve its fundamental properties (volume conservation and the fact that the sum of the three order parameters remains equal to 1)? An answer is given in the next paragraph.

Transport Term in the Cahn–Hilliard System When the Velocity Is Not Divergence Free.

In this paragraph, we are interested in the form of the transport term in the Cahn–Hilliard equation when the advection velocity, denoted by u* is not divergence free but satisfy the boundary condition un = 0 on Γ.

Preserving properties of the Cahn–Hilliard when the advective velocity is not divergence free may be useful in other contexts. For instance, when using an incremental projection method (cf. Ref. [16, 17]), the end step velocity is not divergence free.

The transport term may be written in the conservative or non conservative form (these two forms are not equivalent since a priori div (u*) ≠ 0):

  • nonconservative form: u* · ∇ci,

  • conservative form: div (ciu*).

The conservative form ensures the total volume conservation of each phase (as un = 0 on Γ). This is not the case for the nonconservative form as a priori equation image. Conversely, when using the conservative form, a necessary condition to ensure that the sum of the three-order parameters ci remains constant equal to 1, is div (u*) = 0. Neither the conservative form nor the nonconservative form ensures both volume conservation and the fact that the sum of the three-order parameters remains equal to 1.

We propose to use the following formulation:

equation image

where αi is a constant. This formulation allows to ensure the two desired properties if equation image. To guarantee the consistency with diphasic model, the constant αi may be zero when the phase i is not present. In the sequel, we propose to choose:

equation image

This formulation allows to use an advective velocity which is not divergence free. The term −αidiv (u*) is added in the Cahn–Hilliard system, its role is to re-equilibrate the values of each order parameters to ensure the fact that their sum remains equal to 1. We prove in Section V that this term is of order O(h + Δt), so it does not disturb the consistency of the scheme.

Owing to this formulation of the transport term, it seems natural to adopt the following definition for the capillary forces term in the Navier–Stokes equation:

equation image

This is equivalent to modify the definition of the pressure by subtracting the term equation image.

Time Discretization of the Cahn–Hilliard/Navier–Stokes System.

Finally, the different considerations presented in the previous paragraphs lead to propose the following scheme:

Problem 1.

  1. Step 1: resolution of the Cahn–Hilliard system

    Find equation image such that, for i = 1−3,

    equation image

    with αj a constant: equation image.

  2. Step 2: resolution of the Navier–Stokes system

    Find equation image such that,

    equation image

    where ηn+1 = η(cmath image) and ϱ = ϱ(c), for ℓ = n and ℓ = n + 1.

B. Space Discretization

For the space discretization, we use the Galerkin method and the finite elements method. Let equation image, equation image, equation image, and equation image be finite elements approximation spaces of equation image, equation image, equation image, and equation image, respectively. As the velocity satisfies homogeneous Dirichlet boundary conditions on Γ, we define the following approximation space:

equation image

To simplify the notation, we introduce also the following space:

equation image

We require some standard assumptions on approximation spaces:

equation image(2.3)
equation image(2.4)

Moreover, we assume that equation image, equation image and that the L2-projection equation image (resp. equation image) on the approximation space equation image (resp. equation image) is stable in H1, that is, there exists a positive constant C independent of h such that:

equation image(2.5)

We assume that the approximation space for order parameters satisfies an inverse inequality: there exists a function Cinv of h such that

equation image(2.6)

This property is (for instance) satisfied when the mesh family is quasiuniform, and the approximation spaces are associated to corresponding Lagrange finite elements; in this case, we can choose Cinv (h) = C(1 + |ln(h)|) for d = 2 and Cinv (h) = Ch−1 for d = 3 where C is a constant, which only depends on the mesh regularity (cf. Ref. [18, 4.5.11 (p. 112) and 4.9.2 (p. 123)]).

Finally, we assume that the approximation spaces for velocity and pressure satisfy the so-called uniform inf–sup condition: there exists a positive constant Θ (independent of h) such that

equation image(2.7)

We begin with the definition of discrete functions equation image and equation image at the initial time satisfying:

equation image(2.8)

These discrete functions cmath image and umath image can be obtained from initial conditions c0 and u0 by H1(Ω) projection, or as it is the case in practice, by finite elements interpolation provided that ci0 and u0 are smooth enough.

Assume now that equation image and equation image are given, the Galerkin approximation of Problem 1 reads:

Problem 2 (Formulation with three-order parameters).

  1. Step 1: resolution of the Cahn–Hilliard system

    Find equation image s.t. equation image, equation image, for i = 1,2,3,

    equation image

    where αjh is the constant defined by equation image.

  2. Step 2: resolution of the Navier–Stokes equation

    Find equation image such that equation image, equation image,

    equation image

    where ηmath image = η(cmath image) and ϱmath image = ϱ(cmath image), for ℓ = n and ℓ = n + 1.

Remark 2. For the resolution of the Cahn–Hilliard system, it is equivalent to only solve the equations satisfied by (cmath image, cmath image, μmath image, μmath image) and then to deduce the unknowns (cmath image, μmath image) using the following relationships:

equation image

The resolution of the Navier–Stokes system remains unchanged (cf. Problem 2). In the sequel, in systems where only the unknowns (cmath image, μmath image, cmath image, μmath image) are present, the notationcmath image stands for the vector (cmath image, cmath image, 1 − cmath imagecmath image).

III. UNCONDITIONAL STABILITY OF THE SCHEME

We prove in this section the energy equality which ensures the unconditional stability of the scheme.

Proposition 3.1 (Discrete energy equality). Let equation image and equation image. Assume that there exists a solution (cmath image, μmath image, umath image, pmath image) of Problem 2. Then, we have the following equality:

equation image(3.1)

wheredF(·,·) is the vector (dmath image(·,·))i=1,2,3 and

equation image(3.2)

Proof. The key point of the proof is the following observation: the Cahn–Hilliard and Navier–Stokes systems can be rewritten using the function u* defined by (3.2). Then, standard estimations for the Cahn–Hilliard and Navier–Stokes systems are done (Steps 1 and 3) and an estimation on the L 2 norm of u* gives the conclusion (Step 2).

  1. Step 1: Owing to the definition (3.2) of the function u*, we observe that the system (2.9) can be rewritten as follows:

    equation image

    We take νmath image = μmath image and equation image as test functions in this system. After some standard calculation (see Ref. [9]), this yields:

    equation image(3.3)
  2. Step 2: It is possible to obtain an estimation of the first term of the right-hand side of the previous equality. By definition of u*, we have equation image. Multiplying by equation image, and integrating on Ω, yields:

    equation image(3.3)
  3. Step 3: The system (2.10) can also be rewritten using the function u*:

    equation image

We take νmath image = umath image and νmath image = pmath image as test functions in this system. This yields:

equation image(3.5)

The conclusion is obtained by summing up Eqs. (3.3)(3.5). □

Remark 3. An important difference with the work presented in Ref. [15] in the case of a homogeneous diphasic Cahn–Hilliard/Navier–Stokes model is that no condition is required on the time step to ensure the stability.

The previous stability result enables to prove the existence of solutions for the nonlinear approximate Problem 2.

Theorem 3.2. Given equation image, equation image, we assume that

  • the coefficients1, Σ2, Σ3) satisfy (1.4), the mobility satisfies (1.6), and the Cahn–Hilliard potential F satisfies (A),

  • the discretization of nonlinear termsdF satisfies (2.2) and the following property: there exists equation image (eventually depending on cmath image) such that

    equation image(3.6)

Then, there exists at least one solution (cmath image, μmath image, umath image, pmath image) to Problem 2.

The proof relies on the following lemma from the topological degree theory [19].

Lemma 3.3 (Topological degree). Let W be a finite dimensional vector space, G be a continuous function from W to W. Assume that there exists a continuous function H from W × [0; 1] to W satisfying

  • i.H (·, 1) = G and H (·, 0) is affine,
  • ii.R > 0 s.t. equation image, if H (w, δ) = 0 then |w|W < R,
  • iiithe equation H (w, 0) = 0 has a solution wW.

Then, there exists at least one solution wW such that G(w) = 0 and |w|W < R.

The idea is to link the nonlinear discrete problem to a more simple (linear) problem (using an homotopy, function H of Lemma 3.3) for which we are able to prove existence of solutions (assumption (ii) of Lemma 3.3). The topological degree theory allows to deduce the existence of solutions for the nonlinear problem from a priori estimates, which are in our case deduced from the energy equality (3.1) proved in Proposition 3.1. Such a methodology was used for the approximation of the triphasic Cahn–Hilliard model in Ref. [9]. We only give here the main steps of the proof.

α) Problem 2 is reformulated to enter in the framework of Lemma 3.3. Let W be a finite dimensional vector space equation image. We define a norm on W: for all w = (c1h, c2h, μ1h, μ2h, uh, ph) ∈ W,

equation image

and we introduce the function H such that

equation image

where equation image and equation image, (resp. equation image and equation image, resp. equation image, resp. equation image) are defined with their coordinates in the finite elements basis equation image (resp. equation image, resp. equation image, resp. equation image) of equation image (resp. equation image, resp. equation image, resp. equation image):

equation image

with Mmath image = M0((1 − δα)cmath image +δαcmath image), ϱmath image = ϱ((1 − δ)cmath imagecmath image) for ℓ = n or ℓ = n + 1 and ηmath image = η((1 − δ)cmath imagecmath image). The function G is defined by G: wW → H (w, 1) ∈ W. The problem “Find wn+1 such that G (wn+1) = 0” is equivalent (by definition of the function H) to Problem 2. To prove the theorem, we are going to prove that the functions H and G satisfy the assumptions of Lemma 3.3. The continuity of the function H is obtained using the continuity of the different non linear functions (Dmath image, ϱ and η) and the Lebesgue's theorem. The function H (ℓ, 0) is clearly affine by construction. β) Let (wn+1, δ) ∈ W × [0; 1] such that H (wn+1, δ) = 0. Note that H (wn+1, δ) = 0 is equivalent to say that wn+1 = (cmath image, cmath image, μmath image, μmath image, umath imagepmath image) is a solution of a problem closely related to Problem 2. The same calculations as in the proof of Proposition 3.1 allows to prove the following estimate:

equation image

where equation image. Using Proposition 1.1, the fact that F is nonnegative, the positive lower bounds ϱmin and ηmin for the density and viscosity, the fact that the mobility is bounded from below, the Korn lemma (cf. Ref. [12, lemma VII.3.5]) and assumption (3.6), we can readily derive the following estimates

equation image(3.7)

where equation image is a constant independent of δ and wn +1. The bound on pressure is obtained using the bound on the velocity (3.7) and the inf–sup condition (2.7), which ensures (cf. Ref. [18, 21.5.10, p. 344]) that there exists equation image such that

equation image

Thus, taking νmath image = vh in the system associated with H(wn+1, δ) = 0 enables to bound the L2 norm of the pressure:

equation image(3.8)

where equation image is a constant.

Combining (3.7) and (3.8), we obtain a positive constant equation image independent of δ and cmath image such that

equation image

Hence, taking equation image guarantees that for all (w, δ) ∈ W × [0; 1], H(w,δ) = 0⇒|w|W < R.

γ) It remains to prove the existence of a solution to the linear problem H(wn+1, 0) = 0. This problem can be written as three problems, which are totally uncoupled and the existence of solutions for each of these problems is readily obtained (using inf–sup condition).

This concludes the proof of the existence of approximate solutions.

IV. NUMERICAL EXPERIMENTS

In this section, we provide 2D numerical simulations to illustrate the unconditional stability stated in Proposition 3.1.

The space discretization is performed on square local adaptive refined meshes using:

  • equation image Lagrange finite element for the order parameters c1, c2, c3, the chemical potentials μ1, μ2, μ3 and for the pressure p,

  • equation image Lagrange finite elements for each component of the velocity u.

The adaptation procedures are based on conforming multilevel finite element approximation spaces that are built by refinement or unrefinement of the finite element basis functions instead of cells. All the details about this method and also various examples (in particular, simulations using the Cahn–Hilliard model considered in this article) are described in Ref. [20]. The refinement criterion used in those (un-)refinement procedures imposes the value of the smaller diameter hmin of a cell and ensures that refined areas are located in the neighborhood of the interfaces (i.e., where no order parameter is equal to one). We do not give more details on spatial discretization issues here, as the main goal of this article is to investigate the properties of time discretization schemes.

We compare the results obtained with the unconditionally stable scheme (denoted Uncond. in the sequel) proposed in this article (see Problem 2) and the scheme (denoted Stand. in the sequel) used in Refs. [11, 15] which is obtained using an explicit velocity in the Cahn–Hilliard equation (see “Coupling Terms” section).

The (nonlinear) Cahn–Hilliard system is solved using the Newton algorithm and the Navier–Stokes system is solved using the Augmented-Lagrangian method. All intermediates linear system are solved using direct solvers.

The practical implementation has been performed using the software object-oriented component library PELICANS [21], developed at the “Institut de Radioprotection et de Sûreté Nucléaire (IRSN)” and distributed under the CeCILL-C license agreement (an adaptation of LGPL to the French law).

A. Droplet Oscillation

The first example is a two-phase flow simulation of the oscillations of a 2D droplet due to surface tension. This test case was already used in several articles, see for instance Refs. [22–25]. The initial configuration is a 2D droplet with a perturbed radius: r = r0(1 +α cos(2θ)) (in polar coordinates). More precisely, we choose the square] − 4r0,4r0[2 as computational domain and initialize the order parameters c1 and c2 with the following formula:

equation image

where the interface width ε is given by equation image. We use the values r0 = 0.1 and α = 0.05 in all our simulations.

The perturbed droplet is initially at rest and the only external force is the surface tension σ = 1 (i.e., there is no gravity g = 0). We assume that the two phases have the same densities ϱ1 = ϱ2 = 1 and the same small viscosities η1 = η2 = 10−4. We take a small constant mobility M0 = 10−5 and perform simulations until the final time T = 0.2. The space discretization is fixed: equation image.

Figure 1 shows the time evolution of the kinetic energy (on the left) and of the total energy (on the right) using the Uncond. scheme for different values of the time step Δt.

Figure 1.

Time evolution of kinetic and total energy using the Uncond. scheme for different time steps.

The Uncond. scheme ensures the decrease of the total energy for all time steps. This is not the case when using Stand. scheme. Figure 2 shows a comparison of the energies evolution between Uncond. scheme and Stand. scheme. For Δt = 10−4, the results are very similar but for Δt = 2 × 10−4 or greater, the Stand. scheme leads to a blow up of kinetic and total energies.

Figure 2.

Time evolution of kinetic and total energy using the Uncond. scheme and the Stand. scheme for different time steps.

Figure 3 shows the interface shape and the streamlines (of velocity) at t = 0.04 that we obtain when using the Uncond. scheme (on the left) and the Stand. scheme (on the right). These pictures show 20 contour levels of the order parameter c1 between 0.4 and 0.6, and 50 contour levels of the streamline function. It appears small instabilities in the neighborhood of the interface when using the Stand. scheme.

Figure 3.

Contour level of the order parameter c1 and of the velocity (streamlines) at t = 0.04 using the two different schemes.

B. Two-Dimensional Three-Phase Flow

The second example is a three-phase flows simulation of a gas bubble rising in a liquid column under gravity. Physical properties of the three phases and initial configuration are given in Fig. 4. The interface width is given by equation image and we choose a degenerate mobility, that is, the mobility vanishes in pure phases: M0(c) = 10−5(1 − c1)2(1 − c2)2(1 − c3)2. We take Δt = 5 × 10−5 and equation image for this simulation.

Figure 4.

Physical parameters and initial configuration of test case.

The bubble rises in the heavy liquid (phase ****), penetrates the liquid–liquid interface and then rises in the light liquid (phase ***) entraining some quantity of the heavy liquid in the upper phase. This time evolution is shown in Fig. 5 where simulations performed with the Uncond. scheme (on the left) and the Stand. scheme (on the right) are compared. These pictures show 50 contour levels of the order parameters c1 (in red) and c2 (in blue) between 0.4 and 0.6 (be careful the contour levels of c1 and c2 may coincide).

Figure 5.

Time evolution of the system with the Uncond. scheme on the left and with the Stand. scheme on the right. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

The results we obtain with the two schemes are close: the bubble rises with the same velocity in the two cases. However, we observe two main differences: the first is the form of the bubble (see for instance picture at t = 0.6), and the second is the time at which the column of entrained fluid break up (see pictures at time t = 0.75 and 0.9). These differences are certainly due to the fact that the stabilization modifies the value of the mobility in the Cahn–Hilliard equation and consequently the Uncond. scheme involves a more important diffusion at the interface than the Stand. scheme. Nevertheless, note that in the right picture at t = 0.9, the thickness of the entrained liquid column is equal to the mesh size, so the column is near to break up.

V. CONVERGENCE OF APPROXIMATE SOLUTIONS IN THE HOMOGENEOUS CASE

In this section, we assume that ϱ1 = ϱ2 = ϱ3 = ϱ0 > 0. This implies that the function ϱ(c) is constant: equation image.

The existence of solutions is given by Theorem 3.2. For all equation image, we can introduce the following piecewise constant and piecewise linear interpolations in time:

equation image

Note that, in the sequel, cmath image, μmath image, and umath image always designate the functions of the time variable t defined above, which are naturally indexed with the size of the time discretization. To avoid confusion, we designate the particular values of cmath image, μmath image, and umath image for n = N with the notation cmath image, μmath image, and umath image. The convergence result is the following:

Theorem 5.1 (Convergence theorem). We assume that assumptions of Theorem 3.2 are satisfied, so that a solution (cmath image, μmath image, umath image, pmath image) of Problem 2 exists for all equation image and for all h > 0. We assume that equation image, that the consistency property (2.1) is satisfied and that there exist two constants C > 0 and Δt0 > 0 such that for all Δt ≤ Δt0 and for all n ∈ [[0; N − 1]],

equation image(5.1)

Consider the problem (1.8), the initial conditions (1.11) and the boundary conditions (1.9)-(1.10). Then, for all tf > 0, there exists a weak solution (c, μ, u, p) defined on [0, tf[ such that

equation image

Moreover, for all sequences equation image and equation image satisfying the following properties:

  • equation image and equation image,

  • there exists a constant A (indep. of K) s.t.: (recall that Cinv is given by (2.6))

    equation image(5.2)

the sequences equation image, equation image and equation image satisfy, up to a subsequence, the following convergences when K→ + ∞:

equation image

Remark 4. The assumption (5.1) is obtained in practice by applying Proposition 3.1 and by bounding the term:

equation image

in the right-hand side of (3.1). The way to obtain this bound depends on the scheme Dmath image(cn, cn+1) chosen for the nonlinear terms of the Cahn–Hilliard system. This was largely discussed in Ref. [9].

Remark 5. In the statement of Theorem 5.1, the inequality (5.2) is not a stability condition. When using a quasiuniform mesh family and the associated Lagrange finite elements in 2D, this condition is not restrictive, because we can choose Cinv (h) = C(1 + |ln(h)|). In 3D, it means that to obtain convergence toward weak solutions of the continuous problem, it is necessary that the time step goes to zero faster than the mesh size.

The proof of Theorem 5.1 is inspired from Refs. [14, 15] which deal with the homogeneous diphasic Cahn–Hilliard/Navier–Stokes system. Excluding the fact that we are interesting in a triphasic model, the major difference with these works is the taking into account of the transport term in the Cahn–Hilliard equation. We have to prove that the additional term do not disturb the consistency.

Basically, the proof of Theorem 5.1 is split in three steps:

  • first, the energy equality (5.1) allows to prove that the sequences equation image, equation image, and equation image are bounded in some suitable norms.

  • it is then possible to apply compactness theorems to extract some convergent subsequences.

  • the third step consists in proving that the obtained limit is a weak solution of the system (1.8).

We give the details for each of these three steps below. In the sequel (subsections 5.1–5.3 in Sections V), we assume that assumptions of Theorem 5.1 are satisfied and in particular the notation cmath image, μmath image, umath image, pmath image… denote solutions of the discrete Problem 2, and equation image, equation image, equation image the associated sequences.

A. Bounds on Discrete Solution

In this section, we assume that K is fixed and to simplify notation we omit the index K in the notation hK and NK. The first estimates stated in Proposition 5.2 are directly derived from the energy estimate (5.1).

Proposition 5.2. We have the following bounds:

equation image(5.3)
equation image(5.4)
equation image(5.5)

where K1, K2, and K3 are three constants independent of Δt and h.

Proof. This proof is very similar to the one of Proposition 4.2 in Ref. [9]. Nevertheless, note that it use additional ingredients (Korn lemma (cf. Ref. [12, lemma VII.3.5]), the lower bound for the viscosity η(c) and the fact that the density is constant) to deal with the terms that involve the velocity. □

To pass to the limit in nonlinear equations (cf. subsection 5.3 in Section V), we need strong convergence of the subsequences. For this reason, it is useful to obtain more accurate estimates.

Proposition 5.3. There exist two constants K4 and K5 independent of h and Δt such that:

equation image(5.6)
equation image(5.7)

Proof.

  • i.The estimate (5.6) is obtained from the first equation of the Cahn–Hilliard system.(α) Consider equation image. The first equation of (2.9) reads:
    equation image
    Thus, the inverse inequality (2.6) yields:
    equation image
    Finally, thanks to (5.2) and (5.3), we obtained that there exists a constant K (independent of h and Δt) such that:
    equation image(5.8)
    We are now going to use this intermediate inequality to prove (5.6).(β) Let ν ∈ H1(Ω). Let νmath image be the L2 projection of ν on equation image. Owing to (2.5), we have:
    equation image
    Thus, using (5.8), we obtain
    equation image
    As this inequality is true for all ν ∈ H1(Ω), we have
    equation image
    Consequently, using (5.4) yields:
    equation image(5.9)
    (γ) We now take νmath image = Δt(cmath imagecmath image) in (5.8). This yields:
    equation image
    and so, using (5.4) and (5.5), we have:
    equation image(5.10)
    The inequality (5.6) is readily deduced from Eqs. (5.9) and (5.10) by defining the constant equation image.
  • ii.To obtain estimate (5.7), we begin with bounding the term: equation image for n ∈ [[0;Ni − 1]]. We choose equation image such that
    equation image(5.11)
    as a test function in (2.10) and we sum up the equations to obtain:
    equation image

We then separately estimate each term of this inequality. For term T1, by using the Hölder inequality and an interpolation inequality, we obtain:

equation image

Using the bound (5.3) and the Young inequality yields:

equation image

We conclude using the Hölder inequality and the bound (5.4):

equation image

The term T2 is bounded in the same way:

equation image

For the viscous term T3, we derive the following estimate:

equation image

It remains the terms T4 and T5:

equation image

and

equation image

Finally, we obtain the following result: there exists a positive constant K such that, for all equation image satisfying (5.11), we have

equation image

In particular, for νmath image = umath imageumath image (which satisfies (5.11) owing to (2.10)), we find

equation image

Thus, we obtain

equation image

This leads to the conclusion with equation image. □

B. Compactness Argument, Convergence of Subsequences

The estimates proved in Subsection A in Section V (Propositions 5.2 and 5.3), allow to obtain (up to subsequences) the convergence of the sequences: equation image, equation image, equation image, equation image, equation image, equation image, and equation image. The following propositions give the spaces in which these convergences hold.

Proposition 5.4. Up to subsequences, we have the following convergences when K → +∞:

equation image(5.12)
equation image(5.13)
equation image(5.14)
equation image(5.15)

Proof. The convergences (5.12)–(5.15) are direct consequences of Proposition 5.2. Indeed, it is easy to verify that the estimates stated in this proposition prove that the sequences equation image, equation image, equation image, and equation imageare, respectively, bounded in the following norm: L(0, tf, (H1(Ω))3), L2(0, tf, (H1(Ω))3), L2(0, tf, (H1(Ω))′), and L2(0, tf, (H1(Ω))d).□

The weak convergences we write above are not sufficient to pass to the limit in the nonlinear terms of the Cahn–Hilliard and Navier–Stokes systems. We prove in the next two propositions that it is possible to obtain strong convergence for order parameters and velocity in some suitable function spaces.

Proposition 5.5. Up to subsequences, we have the following convergences when K → +∞:

equation image(5.16)
equation image(5.17)

Proof. The sequence equation image is bounded in L(0, tf, (H1(Ω))3) and its time derivative equation image is bounded in L2(0, tf, (H1(Ω))′). We obtain the strong convergence (5.16) of order parameters by applying the Aubin–Lions–Simon compactness theorem [26]. From this convergence and using the inequality (5.5), we deduce the strong convergences (5.17). □

To prove the result of strong convergence on the velocity, we need to apply a more precise compactness result, because we do not have any estimate on its time derivative. We apply a compactness theorem due to Simon in which the condition on the time derivative is replaced by an estimation on time translates.

First, we write the term to estimate. This term is defined from the discrete function umath image which is piecewise constant (in time) and its time translate. We link this term to the values umath image of the function on each time intervals to exploit estimates proved in Subsection A in Section V. To simplify the notation, we omit in this lemma, the index K in the notation hK and NK.

Lemma 5.6. Let τ ∈]0, tf[. We denote by i ∈ [[0; N − 1]] the unique index such that ti ≤ τ < ti+1. Then, we have:

  • iif τ < Δt then
    equation image
  • iiin all cases, we have:
    equation image

We can now state the proposition giving the strong convergence for the velocity.

Proposition 5.7. Up to subsequences, we have the following convergences when K → +∞:

equation image(5.18)

Proof. The proof relies on a compactness theorem due to Simon [26, Theorem 5, p.84] which state that the embedding

equation image

is compact. The Nikolskii space equation image is defined by:

equation image

with the norm

equation image

Thus, since the sequence equation image is bounded in the spaces L2(]0, tf[,(H1(Ω))d) and L2(]0, tf[,(L2(Ω))d) (cf. Eqs. (5.3) and (5.4)), it is sufficient to prove that it is bounded in the space equation image, to obtain the conclusion. Let τ ∈]0, tf[. We still omit the index K in the notation hK and NK.

  • i.If τ < Δt then owing to Lemma 5.6, we have:
    equation image
  • ii.If τ ≥ Δt then owing to Lemma 5.6, and then using the inequality (5.7), we have:
    equation image
    since we have ti ≤ τ and ti+1 = ti + Δt ≤ 2τ.

In all cases, we obtained the existence of a positive constant K6 (independent of h and Δt) such that:

equation image

This concludes the proof of the convergence of umath image. The convergences of equation image and equation image are then obtained thanks to the inequality (5.5). □

C. Passing to the Limit in the Scheme

The convergences obtained in Subsection B in Section V allow to pass to the limit in the discrete system.

For the Cahn–Hilliard system (without the transport term), this work was already done in details in Ref. [9]. We focus here on the transport term and on the Navier–Stokes equation.

To simplify the notation, we still omit the index K in the notation NK and hK but when we say “convergence” it means K → +∞ (and consequently NK → +∞ and hK → 0).

Transport Term in the Cahn–Hilliard Equation.

Let equation image a given function and equation image. We define νmath image as the H1 projection of the function νμ on equation image. We have to prove the following convergence:

equation image(5.19)

We proceed in two steps, separately considering two terms of the left-hand side: the standard transport term and the additional term that ensures the unconditional stability.

The following inequalities allows to identify the limit of the first term:

equation image

as cmath image is bounded in L2(0, tf, H1(Ω)), umath image is bounded in L2(0, tf, (H1(Ω))d), cmath image (strongly) converges in L2(0, tf, L2(Ω)) toward ci (cf. Eq. (5.17)), umath image (strongly) converges in L2(0, tf, (L2(Ω))d) toward u (cf. Eq. (5.18)) and, owing to assumption (2.3), equation image.

We now use the fact than the sequences cmath image are μmath image are, respectively, bounded in L(0, tf, H1(Ω)) and L2(0, tf, H1(Ω)) norm, the inverse inequality (2.6) and the condition (5.2) on the sequences hK and NK to show that the second term convergences toward 0:

equation image

Thus, we proved that the convergence (5.19) holds. Reusing (exactly as it is) the reasoning presented in Ref. [9] allows to pass to the limit in the other terms of the Cahn–Hilliard system.

Navier–Stokes System.

Let equation image satisfying div (νu) = 0 and equation image such that τ(tf) = 0.

We introduce the space

equation image

The inf–sup condition (2.7) implies that the function νu ∈Hmath image(Ω) which is divergence free can be “well approximated” with functions in Zh. This is detailed in Proposition 5.8.

Proposition 5.8 (Approximation of divergence free functions, [18, eq. 12.5.17]). We have the following inequality:

equation image

Let νmath image be the H1 projection of νu on the space Zh. Proposition 5.8 and the assumption (2.4) show that

equation image(5.20)

We use νmath image as a test function in the first equation of (2.10). We then multiply by τ(t), t ∈]tn, tn+1[, integrate between tn and tn+1, and sum up for n from 0 to N − 1, so that we rebuilt a variational formulation on]0, tf[ × Ω. The unsteady term is modified by a discrete integration by part:

equation image

Thus, we obtain the following formulation of the scheme in which we can pass to the limit:

equation image

The limit of the term T1 is readily obtained from strong convergences (5.18), (5.20) and those of functions equation image toward τ′ in L2(0, tf) (obtained for instance with dominated convergence theorem since the function τ is in equation image):

equation image

The term T2 allows to show that u satisfies the initial condition (1.11) in a weak sense. The convergences (2.8), (5.20) and the uniform convergence on [0, tf] of the function t ↦ τ(Δt(t − 1)) toward the constant function equal to τ(0) yields:

equation image

Concerning the term T3, the following inequality allows to conclude:

equation image

Indeed, as the sequences (umath image) and equation image are bounded in L2(0, tf, (H1(Ω))d), the convergences (5.18) and (5.20) show that the first two terms of the above right-hand side tend to 0. The last one (the term involving the integral) also tends to 0 by weak convergence of equation image toward ∇u in L2(0, tf, (L2(Ω))d) (a component-by-component reasoning gives the result, as for all 1 ≤ i, jd, the function (t, x) ↦ ui(xmath image(x)τ(t) lies in L2(0, tf, L2(Ω))).

The term T4 is treated in the same way:

equation image

the conclusion being now obtained using the convergences (5.18), (5.20) and the fact that the two sequences umath image and equation image are bounded in L2(0, tf, (H1(Ω))d).

The limit of the term T5 is obtained using the following convergence (up to a subsequence):

equation image(5.21)

This convergence is proved using the dominated convergence theorem (the viscosity η is a bounded continuous function and equation image strongly converge in L2(0, tf, (L2(Ω))3), almost everywhere up to a subsequence).

Thus, using (5.20), (5.21), the fact that the sequence equation image is bounded in L2(0, tf, (H1(Ω))d), and the weak convergence of equation image toward Du in L2(0, tf, (L2(Ω))d), we obtain

equation image

By (5.20), the convergence of term T6 is straightforward:

equation image

The convergence of the capillary forces term T7 is obtained as follows:

equation image

The first two terms of the right-hand side tend to 0 thanks to convergences (5.17) and (5.20), as the sequences (cmath image) and (μmath image) are bounded in L2(0, tf, H1(Ω)) and L(0, tf, H1(Ω)), respectively. The last term tends to 0 by weak convergence of ∇μmath image towards ∇μj in L2(0, tf, (L2(Ω))d).

Finally, it only remains to prove that the residual term T8 tends to 0. This simply comes from the fact that:

equation image

and

equation image

In conclusion, we proved that:

equation image

To finish, passing to the limit in the constraint equation yields:

equation image

VI. CONCLUSIONS

We proposed in this article an original scheme for the discretization of a triphasic Cahn–Hilliard/Navier–Stokes model.

This scheme is unconditionally stable and preserves, at the discrete level, the main properties of the model, that is the volume conservation and the fact that the sum of the three-order parameters remains equal to 1 during the time evolution.

We proved the existence of at least one solution of the discrete problem and, in the homogeneous case (i.e., three phases with the same densities), we proved the convergence of discrete solutions toward a weak solution of the model (whose existence is proved in the same time).

The main perspective is the study of the convergence in the case where the three fluids in presence have different densities. Even if the energy estimate (and the existence of discrete solutions) are still true in this case, it is delicate to obtain sufficient estimates which would lead, by compactness, to strong convergence on the velocity which is necessary to pass to the limit in nonlinear terms. Indeed, the Navier–Stokes equation involves a term of the form utϱ. The time derivative of the density is not very smooth, because it is a function of order parameters whose time derivative is in L2(0, tf, (H1(Ω))′).

Acknowledgements

The author expresses his gratitude to Franck Boyer and Bruno Piar for their support during the preparation of this work and thanks the referees for their careful reading of the article.

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